Nuclear relaxation and electronic correlations in quasi-one-dimensional organic conductors. I. Scaling theory

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Nuclear relaxation and electronic correlations in quasi-one-dimensional organic conductors. I. Scaling

theory

C. Bourbonnais

To cite this version:

C. Bourbonnais. Nuclear relaxation and electronic correlations in quasi-one-dimensional organic conductors. I. Scaling theory. Journal de Physique I, EDP Sciences, 1993, 3 (1), pp.143-169.

�10.1051/jp1:1993122�. �jpa-00246706�

J.

Phys.

I France 3 (1993) 143-169 JANUARY1993, PAGE 143

Classification

Physics

Abstracts

76.60E 74.70K 75.40E 75.50E

Nuclear relaxation and electronic correlations in quasi-one-

dimensional organic conductors. I. Scaling theory

C. Bourbonnais

Centre de Recherche _en Physique du Solide,

Ddpartement

de Physique, Universitd de Sherbrooke, Sherbrooke, Qudbec, Canada, JIK-2Rl

and

Laboratoire de

Physique

des Solides, Universitd de Paris-S~ld, Bit. 5ID, 91405

Orsay

Cedex, France

(Received 21 April 1992, accepted in final

form

4 September 1992)

Abstract. In this paper the results of the scaling theory for the

quasi-one-dimensional

^electron

gas model _are used _to make _a detailed

analysis

of the temperature ^{variation of the nuclear} relaxation rate for

quasi-one-dimensional

conductors which present an

antiferromagnetic

critical point. ^{From the extended} dynamic scaling

hypothesis,

we show how the statics, the

dynamics

and the

dimensionality

of

antiferromagnetic

and uniform spin fluctuations _are involved in the power law behaviours of

Tj'

for both the normal and the critical temperature ^{domains. The full}

expressions

of the

antiferromagnetic

contribution _to Ti~, the related critical indices _as well _as the

dimensionality

crossover

scaling

functions _are derived in the _cases where either the interchain

exchange

_or the

quasi-

ID

nesting

of the Fermi surface drives the transition. As for the influence of uniform

spin

fluctuations, _a derivation for the temperature

dependent

enhancement of the

magnetic susceptibility x~(T)

in _one dimension is

given.

It is found that the harmonic character of

collisionless paramagnons

yields

to the following

scaling

law

Ti~ ~T[Xs(T)]°~~~~

which dominates at

high

temperature ^with an indice that

depends

_on the

dimensionality

D of the system.

All the direct calculations _are shown to be consistent with the scaling

hypothesis.

1. Introduction.

This paper is the first of _a series devoted _to

spin-lattice

nuclear relaxation in

quasi-one-

dimensional

(quasi-lD>

conductors. It focuses _on the

scaling approach

to the temperature

dependence

of the nuclear

spin-lattice

relaxation _rate

(T~ )

_as obtained

by

NMR for correlated

quasi-ID

conductors.

It is

by

far

quite

well established that _{as a} local

probe,

nuclear relaxation _can

play

a

fundamental role in the

understanding

of the

complex

features shown

by spin

fluctuations in

strongly anisotropic

materials systems ^like

weakly coupled spin

chains

Ii, organic

conductors

[2, 3]

and

superconductors [4].

Correlations in

quasi-

ID

organic

conductors and insulators _are

characterized

by

_many

length

_or

temperature

scales. These _are

mainly

associated with short

wavelengths

or

large

wave vectors of the

spin

fluctuations

spectral weight.

For

repulsive

interactions, these

spin

correlations _are

antiferromagnetic (AF)

and their

growth

_as the

temperature

^{is lowered often leads to}

long

_range order. Since AF correlations _are collisionless that

is,

non-diffusive in

character,

their influence on

Tj

will then appear in the temperature

dependence

whereas _a

frequency

variation is

only present

on the scale of the electronic Larmor

frequency

_w~ for a scalar

hyperfine

interaction and

therefore,

it will be

negligeable

for

temperatures

^T » w~. As for the effect of critical AF

ordering,

it

gives

rise _{to a} power law

singularity T~~ (T-T~)~*

_near the Neel temperature

T~ [3, 5].

Seen _{as a} critical

phenomena,

^{this results from} the

divergence

of the AF correlation

length

f at the

approach

of

T~, thereby indicating

that _a

scaling approach

to the temperature ^{variation of}

Tj

should

apply [3]. Actually,

this _can be

readily

illustrated if _one looks at the basic

expression

of

T~

^for a scalar

hyperfine

interaction

namely [6],

Tj

=

2

y( T(gp~)~~ id~q

^{[A~[~ x}

^{] (q, w~)/w~ (h}

⁼

^k~

^{= I}

^{), (1)}

where _x

[ (q,

_w

~ is the

imaginary

part ^of the electronic

susceptibility

for _a direction _transverse to the

applied magnetic

field and at the _{wave vector q} and the nuclear Larmor

frequency

w~. y~ is the nuclear

gyromagnetic

ratio and

A~

^{is the}

hyperfine coupling

constant. From

(I),

^it

is clear that

T[

_can be _{seen as a sum} of two

contributions, Ti [q 0]

and

Ti [q Qol,

related to uniform and

staggered magnetic

correlations

respectively.

In the

large q~oo

integration

sector where

Qo corresponds

to the AF modulation wave vector, one can

apply

the

dynamic scaling hypothesis

to the three

quantities d~q, Xi (q, w~)

and _{w~ so} that

they

_can be dimensioned in _terms of

I

_to

a

given

_power

(exponent)

for each of them

[3, 7].

For the first two

quantities,

the exponents are related _to the statics while for the third _one, it _concems the

dynamics

of

spin

fluctuations. Therefore the essential

pieces

of information

conceming

the

statics,

the

dynamics

_as well _as the

dimensionality

D of

spin

correlations should be in

principle

contained in the

T~

temperature

dependence.

A

quite interesting

consequence of the

anisotropic

character of

quasi-

ID systems ^{consists in} the fact that the

scaling hypothesis

is _not restricted _to the critical domain but it

equally applies

far above

T~

where AF correlations should have _a

purely

ID character. Within the ID

temperature

iomain itself,

different

regimes

of correlations _can

successively

_occur

(e.g.

strong and weak

coupling regimes, etc.) [3].

This is known _to be present for

example,

in the

paramagnetic

temperature ^{domain of the}

sulphur

series

(TMTTF )~X (see

the _next paper of the

series)

and the

sulphur-selenides

series

(TMDTDSF)~X [8]

at low pressure which _are characterized

by

a correlation gap. It follows that AF correlations in

quasi-one-dimensional

systems can be _{seen as a}

problem

with many temperature or

length

scales associated with

different

temperature dependences

of

T~ [q Qo].

In

addition,

these are related to the _so- called _crossover

scaling effects,

which in the

theory

of critical

phenomena

_are known _to lead _to

a

change

in the critical indices of

singular quantities

as the

temperature

^is varied

[7].

Crossover features _can be

analyzed

with the

help

of the extended

scaling hypothesis widely

used in the

theory

^of

anisotropic

critical

phenomena [7].

Its _use is of great ^{interest here since it will}

impose

very

precise

constraints _on the various forms

Tl'[q Qo]

_can take and which therefore must be satisfied

by

_any

microscopic

calculations.

This

approach

tums out to be relevant not

only

for

Tj [q Qo]

but also for the uniform part

Ti [q

0

].

Uniform _{or «}

ferromagnetic

_» correlations _are

non-singular

in

quasi-

ld conductors but from the enhancement of the static and uniform

susceptibility,

their

amplitude

which is

related to paramagnons is _not small and

presents

a temperature

dependence.

Since the statics

N° I NUCLEAR RELAXATION IN ORGANIC CONDUCTORS. 145

and the

dynamical properties

^{of these} paramagnons differ from those at

Qo,

the

temperature dependence

of

T~ [q 0]

is thus

expected

to be also

quite

different.

Direct

microscopic

calculations of

Tj [q Qo]

are

presented

in the section 3 of this paper and

they

will be made

exclusively

in the framework of the

quasi-

ID electron _gas model

[9-

II

].

This model is characterized

by

the small ratio

t~/t~

ml between the transverse and the

longitudinal single

electron

hopping integrals

whereas the electrons

locally

interact

through

the set of

couplings

constants g~ _~

resulting

from the

«

g-ology

_»

parametrization

of the direct electron,electron interaction

[I Il.

The relevance of this model for the

phase diagram

of the

Bechgaard

salts and their

sulphur analogs

will be

extensively

^{discussed in the} next paper

[8, lo, 12].

Its

predictions conceming

the

antiferromagnetic ordering

as a function of

hydrostatic

pressure are found _to be

highly

consistent with the _ones of the observed

phase diagram

lo, 12].

Furthermore, intrachain

many-body

corrections _to

particular

combinations of the

g's

are well known to be connected with either

long wavelength charge

_or

spin

excitations and which _can be related to observable

quantities

like

resistivity

and

magnetic susceptibility [3,

9, II,

13].

As far _as

long

_range order is

concerned, depending

on the ranges of values for the

g's

and t~, ^different

microscopic

mechanisms _can drive the AF

phase

transition. Three different

cases can be considered and each of them will in turn

impose

a distinct temperature ^variation to

T/~[q~oo]

outside the critical domain above

T~.

This

emphasizes again

that relevant

microscopic

details _can be extracted from the temperature ^{variation of nuclear} relaxation.

In the half-filled band _case, ID strong

coupling

effects for the electronic

umklapp

term g~

gives

^rise to a non-zero correlation gap A~ ^for

charge

excitations. At lower energy, electrons

are confined

along

the chains _so that the interchain

single

electron band motion,is frozen and has _no chance to

develop.

Virtual electron transfer is

possible

however, and _a interchain AF

exchange coupling (IEX)

of kinetic

origin

is

generated, thereby assuring,

_{as a}

two-particle microscopic mechanism,

the interchain

propagation

^{of AF} correlations needed for

long

range order

[10].

For weak

umklapp

effects there is _no correlation gap above

T~,

the chains remain metallic down to the

vicinity

of the critical

point

and AF order

acquires

an itinerant character.

Depending

on the

amplitude

of the

g's however,

two situations _{can occur.} On _one hand, ID correlations

though gapless,

_can be

sufficiently important

for the IEX mechanism to be still the

driving

force of the transition

[10].

^{On the} other hand, if the

repulsive g's

are

weak,

there is _a low

temperature

^{domain where the electron and the hole band motion is} no

longer

^confined

along

the chains. The electrons then

undergo

_a

single particle dimensionality

_crossover at a

temperature T~i ^{below which under}

good

2D _or 3D

nesting

conditions for the Fermi surface at

Qo,

an AF

phase

^transition occurs

corresponding

to a nested

antiferromagnet (NAF) [14].

The renormalization group calculations

[10]

of the ratio

Xi (q,

_w

)/w

for the above three different

cases will be used in the limit _w

- 0 and _the

explicit

forms of

T~ ~[q

~

Qol

will be

given.

As for the uniform part

Tj [q

0 which is treated at the end of the section

3,

the absence of

singular

_«

ferromagnetic

_» correlations allows _{us to} consider

only

the contribution of ID paramagnons

neglecting higher

dimensional effects. In the

appendix A,

_we show that their contribution _to the

dynamic susceptibility

_at small q is

purely

harmonic _so that the

resulting expression

for the enhancement of

Ti lq 0]

_can be

uniquely expressed

as a function of the static and uniform

magnetic susceptibility x~(T).

2.

Dynamic scaling

results.

A distinctive feature of the present

problem

with respect to usual critical

phenomena

is the existence of _two different types ^of correlations which,

obviously,

can not be described

by

a

single

critical parameter. Differences _are also

present

in the

dynamics

involved.

Indeed,

«

ferromagnetic

_»

correlations, though

_non

singular,

_are related to _a conserved

quantity

which

is the total

magnetization.

Such _a

quantity

does not exist for AF correlations. It follows that the so-called

dynamical

_{exponent z} is also different for each type ^of correlations

[15],

_a feature that is confirmed

by microscopic

calculations _{as we} will _see.

Finally,

the

spatial

_«

rigidity

_» of the

ferromagnetic

order

parameter

^{is far from}

being

assured in low-dimensional electronic systems since open Fermi surfaces for fermions _{on a} lattice

strongly

favor

nesting properties

at finite

wave vectors.

Looking

at the basic

expression (I)

for

Ti

_{~, we see} that the

integral

_{over q is}

naturally split

into _two parts

namely,

Ti'~Ti~iq~0i+Ti'iq~ooi, (2>

where both the AF

(q Qo)

and the uniform

(q 0)

pans will be treated

separately.

ANTIFERROMAGNETIC _PART. At the

approach

of the AF critical

point,

the relaxation _rate

becomes

singular.

This

singularity

of

Tj'[q~oo]

is the direct consequence of strong

fluctuations in the local

magnetic

field due to correlations of the

staggered magnetization

that become

long

_range as T

-

TN.

As

previously

mentioned in the

introduction,

_{one can}

identify

three

scaling quantities d~q, Xl

and _w. The

dynamic scaling hypothesis [3, 15]

tells _us that

near

TN,

the

only

relevant

temperature dependence

for these

quantities

_can be

expressed

ⁱⁿ

terms of the critical

parameter

^of the AF correlation

length

~AF " ~0

~A/

,

(~)

namely, i~~

=

r'(T TN)/TN

which vanishes _at the transition. Here r' is _a

positive

constant, k _~ 0 is the critical index

(in

the notation Ref.

[7],

the dots refer to

quantities

_near the critical

point).

For _a

quasi-

ID system, the correlation

length

is

expected

to be

strongly anisotropic

and this will appear in the coherence

lengths to

~

»

lo

~,_~ that _are assumed to be

regular

functions of the temperature ^{in the}

neighborhood

of

TN.

Dimensional

analysis

tells _us that each component q, will then scale like

iii,,.

As for the

frequency

_w, it scales like the inverse of _a characteristic

time scale for the relaxation of AF fluctuations

[15, 16].

As _we

approach T~,

correlations become

long

_range and _a critical

slowing

^down takes

place

_so that the time scale for the

relaxation of correlations goes ^to

infinity.

Of _course, in the _context of nuclear relaxation _rate, the electronic _{w~ or} the nuclear _wN Larrnor

frequency

_{can act as a} cut,off for the critical

slowing

down.

w can then be dimensioned _as

flF£~

which vanishes _at

T~.

Here

I ~ 0 is the

dynamical _exponent

and r is _a short range characteristic

frequency

scale which is

regular

at

T~. Finally, Xi (q, w)

scales like _Xi

(q,

_w

)

which in _tum scales like

ij/

near

Qo

and for small _w. Here

j

stands for the critical index for the static

staggered magnetic susceptibility

at

Qo [16].

It follows that

Xi (Q

+

Q0, °')

_"

~A/

d~_[~<

fAF,1,

^°'~

~A/~l, (~)

which is valid for q,

i~~,,

~ l and 0

~ w

f~' ij/"

_~ l, and where 3t is a

scaling

function

[15, 16].

From the above

scaling arguments

we will write

l~l

[~ Q01

_~ C

(Q0) rA/ _(rAF

~ l

,

(5)

where

f _(Q~)

=

2

y( T(A~~(~[r~ f0a fob f0c]~

X

X

d~ (Q f )

3~

[q, f

_AF,

,, W l~

rA/

^~l~ _°' ~

~A/

^~

) (~)

N° I NUCLEAR RELAXATION IN ORGANIC CONDUCTORS. 147

is _a

regular quantity

_near

T~.

^{The critical} index of

Tj [q Qo]

is

given by

d= I-Dk+ik, (7>

and therefore combines

dimensionality,

statics and

dynamics

of AF fluctuations.

For _a

quasi-lD

system

sufficiently

far above the AF critical

-point,

the AF fluctuations

undergo

a

dimensionality

crossover above which correlations must scale

according

to their

purely

lD character with _a related _set of critical indexes

[10].

The _crossover temperature ^will be denoted

by

_T~. How far from

T~,

_T~ takes

place depends

on non-universal _or

microscopic

features of the model

[7, 10] (see

Sect.

3).

Since

purely

lD systems cannot sustain

long

_range

ordering, T~

=

0 and the correlation

length

takes the form

f

_"

to rA/, (8)

where

r~~=T/Eo

and

Eo

is _a

high

_energy cut-off.

to

^is a short

length

scale and

v~0 is the ID critical index for the AF correlation

length

at

Qo= (2k~,0,0),

k~

being

the ID Fermi _{wave vector.}

Repeating

the steps

given

above, _one

immediately

finds the

following

_power law behavior

Tj [q

+ 2

k~]

m C

(2 k~) rj/

,

(9)

where for _a D

= I system, one has

if ₌ y v + zv I,

(lo)

and

C

(2 kF)

" 2

y~

(AQ~

~(1~E0 to

X

id (qfAF)

_d~lD

[~fAF,

°'~

~A/~ II (°'l~ ~A/~

,

II)

for

qf~~

_~ ^l and 0 _{~ w} r

rjj"

~ l. Here C

(2

_k~ is considered _{as a}

temperature independent quantity.

For _a

quasi-

ID system,

T~ [q Qo

should evolve

continuously

from

(9)

to

(5)

_as the temperature ^is

dropped

below T~. ^This raises the

question

of

compatibility

for the above _two

scaling expressions.

Such _a situation is well known in the

theory

of

anisotropic

critical

phenomena. Indeed, according

to the extended

scaling hypothesis [7],

the total

Ti'[q _Qol expression

should take the

following

form

T~ [q Qo]

_m ~

[Bg~ ^/T~~] rj/ (T

_~

T~)

,

(12)

above the _crossover and

Ti iq Qoi

_~

i _{(gi >(T}

_TN

>~ ^~

(iAF

_~ l_>,

(13>

within the critical domain. Here ~ is _{a crossover}

scaling

function of the small

anisotropic

parameter g~ and which for the present

problem

coincides with the interchain

coupling

parameter. It is g~ that _assures the existence of _a finite T~ ^and

T~.

The _crossover temperature ^is

expressed

as

T~

g)~~

_oz TN

,

(14)

where

#~

^{is the so-called} crossover exponent

[7, 10].

It characterizes the way the

change

ⁱⁿ

dimensionality

and exponents ^is achieved. This is also

clearly

illustrated

by

the form the

crossover

scaling

coefficient must take _near

T~ [7]

that

is,

A

(gi

=

Am gl

^~°

*~~~, (15>

where

A~

is _a non-universal constant. In the presence of many crossovers in the ID

regime,

this formula _can be

easily generalized

to

give

J~

( (~)

_"

~lm

~l ^{~~° ~~~~~~~}~2 ^{~~~ ~~~~~~~} ~i ^~~~ ~~~~

,

where the

(g), (if)

and

(~bj'~) corresponds

to sets of ID small parameters,

Tj~

and

crossover

exponents respectively.

Now if _one associates with each g, the characteristic

temperature ^scale T)~J g~~~~~ one can write

~m(I Po

T(2)

^Wi ~m

On

Tj [Qo

_m

A[

^{S S fi}

ij/ ₍₁₆₎

~o

_T(~~

Tfl~

where

again Ii

is _a non-universal _constant. From this

expression,

all ID

temperature

^intervals [T)~~,

T)~~

_~~] contributes the _same way ^to

T[

and this

clearly

reflects _a

scaling _property.

UNIFORM _PART. For

quasi,lD

electronic systems ^with

repulsive interaction,

AF correlations

are

naturally expected

to grow as the temperature ^{is lowered. This} must then _occur at the expense of uniform

spin

fluctuations whose

amplitude

is indeed found to be

monotonically depressed. Furthermore,

the

quite

different

dynamics expected

for the latter indicates that the

related «critical» parameter, say r~, should have _a

completely

different _structure than

r~~ and

i~~. Assuming

the existence of such a « critical _» parameter for q 0

spin fluctuations,

one can

again apply

the

scaling hypothesis

to the _q 0 domain of

integration

of

(I)

with the result

T~~[q~0]mcoTri* ₍₁₇₎

In

complete analogy

with the AF part, ^the uniform _« critical _» index is

given by 3

= y DP ₊ 2P

(18)

and the constant

Co

whose _{structure at q} 0 and _w

- 0 is similar to the one

given

in

(I I),

is considered _as

independent

of the temperature. ^Here y, P and 2,

correspond

to the exponents ^of the static and uniform

susceptibility (Xs rjY),

the correlation

length (f~ _ri ")

and the characteristic relaxation time for uniform fluctuations

(rF

ri

~) respectively.

From

(17),

if

3

~

0,

the uniform relaxation will be enhanced with

respect

to a linear

temperature profile

which

prevails

for non-correlated metals

[6].

Since there is _no apparent

tendency

to

long

_range

ferromagnetic ordering

in the systems ^under

study,

we will _not consider the

scaling

with the

possibility

of _a

dimensionality

crossover.

3. Direct calculations for the nuclear relaxation rate.

3.I CHoicE _{oF THE MODEL.} In this section, we shall be

making

use of the well known

results of

quasi-lD

electron gas model for the

explicit

calculation of

Tj~

in presence of electronic correlation effects. The relevance of this model for the

description

of electronic

N° I NUCLEAR RELAXATION IN ORGANIC CONDUCTORS. ₁₄₉

properties

^{of the}

quasi-

lD conductors first follows from band calculations

[17] together

with

various

experiments

made _on

organic

conductors which support ^the

anisotropic

_sequence

t~~

^~f

t~~

~

between the transverses and the

longitudinal single

electron

hopping amplitudes.

Since _{we are} interested in low energy or

temperature properties,

the electronic energy

spectrum along

the chains is

usually

taken _as linearized around the lD Fermi

points

_±

_k~.

For _a square lattice of

conducting

chains the free electron energy spectrum ^{of the model} Hamiltonian is

given by

e~(k)

₌

v~~pk k~)

2 t~ _~ cos

(k~~ d~ )

2 t~ cos

(k~ d~ ), (19)

where p refers to the

longitudinal right

_~p

= + or left ~p

=

going

electrons. In the

following,

_we will

put

^the interchain distance

d~

= I. Such _a spectrum ^differs

slightly

from the

more elaborated band calculations

[17]

but it does contain the essential characteristics of the electronic band motion for the

Bechgaard

salts and their

sulphur analogs.

As far _as the electron-electron interactions _are

concemed,

the natural

tendency

to AF

ordering

found in these materials suggests ^that

only

^the intrachain part ^{of the} electron-electron interaction needs to be retained. It will be described in the so-called _«

g-ology

_» framework from which _{one can}

identify

four different

coupling

constants,

namely

the backward

(gi ),

the forward

(g~ ),

^{and the}

umklapp (g~ ) scattering

terms between

right

and left

moving electrons,

and

finally

the small

momentum transfer

(g4)

between electrons that

belong

to the _same branch

[9].

In the limit of the Hubbard

model, they

all reduce _{to a}

single

interaction parameter g, =

U.

Perturbation

theory

shows the presence of

singular logarithmic

corrections of the form g~ ^In

(max (2 T,

_v~q,

w)/Eo)

for the

two-particle

vertices

r,

with

Eo

as the band width energy cut-off

[9].

It acts as the ultraviolet

regulator (Eo

²

EF)

of the

perturbation theory.

An

important quantity

that

naturally

_emerges form these

singular

terms is the ratio

f E~/T ⁽²⁰⁾

which is of the order of the

single

fermion coherence

length

and the thermal de

Broglie

wavelength

of _a

single

electron _near the Fermi level

[10]. f

then

gives

the

spatial

_range ^for the

phase

coherence for each

particle

^involved in the vertices and its role is similar _to the _one of the correlation

length

in the

theory

^{of critical}

phenomena [10, 15, 16].

Within

f

^for

example, single

^and

two-particle

correlations _are self-similar with respect to a

change

of

length

_{or energy}

scale. The

logarithmic

terms of the

perturbation theory

for the various relevant

quantities

lead

to similar contributions at each energy interval below

Eo

and this is known _to lead to

homogeneity properties (see

^for

example Eq. (16)).

In the bandwidth cut-off scheme for

example,

the total Hamiltonian of the electronic system ^{with the scaled bandwidth}

Eo-Eo(I)

=

Eoe~~

_with

I

~ 0

keeps

the _same form

except

^for a renormalization of the

coupling

constants g, ^and the electronic

density

of states at the Fermi level. The renormalization of the

g's

due to lD

many-body

effects at different energy scales follows from the well known

recursion formulas

[9,

10,

18]

_:

S(2>~->i)= ³¹¹ -1(2>~->i)1 ^(21b)

~~

=

#3(2 _#~ #i ) ii ⁽²

#~

#i )j ^{#]/4, (21c)}

in the second order of the renormalization group. Here

di

_{is the} infinitesimal

generator

for the

scaled band width

Eo(I).

The

explicit dependence

on the temperature appears

through

^the

boundary

conditions _at

I

= In

(E~/T)

for the solution of

(21).

On observes that the

scaling equation

for g~ is

independent

of the two others this

actually

reflects _a

quite important

property ^{of the model}

namely,

the

separation

between the

long wavelength spin

and

charge

excitations

[9].

As shown in the

appendix A,

_g~ is related to

long wavelength spin degrees

of

freedom while the combination

2g~

_-gi is connected to

charge degrees

of freedom. A

straighforward analysis

of the

umklapp

term g~ shows that it is

uniquely

involved in the

long wavelength charge degrees

of freedom

[9].

^{In the}

repulsive

sector of the

coupling

constants which will be relevant to us that

is,

_gi

~ 0 and gi 2 g~ _~

[g~ (which

includes the

special

case of the

repulsive

Hubbard model

(gi

~,~ ⁼ U

~

0)),

these second order recursion formulas lead _to the

following asymptotic values, gi* -0, gf

- aru~ ^and

gf

^-2 grv~ as I

- cc

(T

- 0 which _means that

charge degrees

of freedom _are characterized

by

_strong

coupling

and the presence of _a correlation gap A~ ^{while the uniform}

spin

excitations remain

gapless [I1, 18].

In the

following,

_we will _assume the

relationship

_A~

m

grT~

^{between the} correlation gap and the temperature T~ ^{at which the} Mott-Hubbard

charge

localization becomes

perceptible [9, 10].

Many-body

corrections _are also present ^{for the}

single panicle density

of _{states at} the Fermi level.

Actually one-particle self-energy

corrections _are also

logarithmic

and in second order of the renormalization group one has the

following scaling equation [18]

_:

$

_in

_{(zi )}

=

(#]

⁺

^{#] #~ #i}

⁺

^{#] (22)}

In the notation of references

[9, 10], zj (i)

refers to the renormalization factor of the _one-

particle

propagator. ^{It also coincides with the ratio N}

[E~, I ]/N (E~)

between the renormalized and the bare

single particle density

of _states per

spin

at the Fermi level

(N (E~)

=

I/grv~).

The solution of

(22)

_can be written _{as a} power

scaling

form

zj i(T> _(T/E~>8

,

(23a>

for g~ =

0,

with o

=

(2

#~ g~

)~/16,

and

zj~(T) _{(T~/E~)~ (T/T~)~*}

,

(23b)

for gi ~ 0 and gi 2 g~ ~ g~

[,

with o

*(g,*

_- 3/4. The absence of

quasi-particles

states at the Fermi level _as T

- 0 K indicates that

only long wavelength spin

and/or

charge

excitations

remain.

They

will contribute _to the retarded lD

dynamic

AF

spin

_response function

xi~(q

+ 2

k~,

_w ^{which is} useful for _our purposes. An

important

related

scaling quantity

is the

auxiliary

response function which is defined via the real part _XID, ^{that is}

[9, [[j,

X _ID ~ ~~F ~X~D/~fl (1~0~X),

(~4)

where

x ₌ max

(2

T, _{v~ q, w}

)

It

obeys

to the

following scaling

relation

[18]

din

kiD

~ l

~i "#2+#3-j (#2+#(-#2#1+j#() (25)

In the limit _w

- 0 for the

imaginary

part X^" relevant to

Tj

_~, one has the

interesting

relation

[10, 19, 20]

xl'~(q

₊ 2

k~,

w,

T)

m

kiD(q

+ 2

k~, T) Xii _(q

₊ 2

k~,

w

), (26)

N° I NUCLEAR RELAXATION IN ORGANIC CONDUCTORS. _lsl

where

Xll(q

_{+ 2}

_kF,

W - 0

=

j dk(n i-

_e~

(k q>i

n

ie~ (k>i

x

x 8

(w

_{2 e~}

(k>

+ v~

q>

- N

(E~) rw/cosh~ _(p

v~

q/4 (27)

is the

imaginary

_part of the bare response function _near

2k~

and _at small _w. Here r

= gr/8 T is _a characteristic time scale for the relaxation of 2 k~ fluctuations.

From the

asymptotic

values of the

coupling

constants in the presence of A~ ^{and for}

I

- cc, it follows

immediately

that

RID

^varies as a power law that

is, RID

x~ Y

,

(28)

with the critical indice _y

= 3/2. This value is known _to be _an overestimation and in

higher

order of the

perturbative

renormalization

procedure

would

bring

_y closer _to

unity. Actually,

other

analytical approaches [9, 21]

in the context of the

repulsive

Hubbard model have shown that for _x below

A~,

one has

y = ,

(29)

which _can be considered _{as an exact} result in the

coupling

sector considered

[21].

Besides the

precise

value of _y, the

scaling equation (23)

allows _to

give

a continuous

description

of the

response function from the weak

coupling T»T~ regime

to the strong

coupling

one

T «

T~.

^If one

neglects transients,

the solution of

(23)

is consistent with the

following scaling

form _:

ij~(q

2

k~,

_w,

T>

=

ij~(Ap/Eo) ix/Api-

^Y

(30>

Here the constant

kiD(A~/Eo)~l

^is a

scaling

coefficient that

gives

the power law

contribution _to

kiD

^above

A~. Neglecting again transients,

_{one can} write

gj~(x>~ (x/Eo>-Y°, (31>

where in the weak

coupling regime,

_one has _yo

m

(#~ #i/2) _(#~ #i/2)~

for g~ =

0,

is 2

non-universal. As for the real

part

^{of the} response

function,

_{one gets}

according

to

(24)

and

(30- 3l)

:

k(D(q

₊ 2

k~,

_w,

T)

m

(grv~ y)~ g~~(A~/Eo) [x/A~]~

^Y + X

(D(A~/Eo), (32)

where

again X(D(A~/Eo) gives

the contribution to

X'

from energy scales above

A~.

^From

(22)

and

(29),

_one has

neglecting

the transients _:

x'(q

₊ 2

k~,

_w,

T>

m

(wv~ yo>-

ⁱ

i(x/Eo>~

^~°

i1 (33>

From the dimensional

analysis

of the above power law argument _x, one finds

iwi

=

iTi

=

iqi

=

ii- ~i. _(34>

From the definition

(8),

it follows that

v ₌ z

=1, (35)

for the coherence

length

and the

dynamical

indexes. Since _w, T, and v~ q

always

enters on the

same

footing

in all

microscopic

calculations, the above values of

exponents

can be considered

as exacts in lD.

An

approximate

and useful form for the q

dependence

of the power laws

(30, 32, 33)

and that will be useful in the direct calculations of

T~

~, is

given by [22]

x ₌ max

[v~

_q, ²

T]

_-

[(v~ q)~

₊ gr~T~]~~~

(36)

3.

Strongly

correlated

quasi.lD antiferromagnets.

Since

purely

ID systems cannot sustain

long

range order at any finite

temperature,

one must

specify

the different

possible

_ways the interchain

coupling

can lead _to the

propagation

of AF correlations in the _transverse direction.

Very simple

_{arguments can} be used to this end and in

particular

for

systems

^like

(TMTTF)2X

and

(TMDTDSF)2X

which

present

a correlation gap.

The _nature of the effective interchain

coupling

when T~ ^«

T~,

_as it is the _case for

(TMTSF)~X,

is known to be

slightly

_more delicate _as we will _see

[10].

An essential

microscopic

_process for the transverse

propagation

of AF order is

provided by

the interchain

single

electron transfer t~.

Owing

to the strong

anisotropy

t~

~ ~

« t~

however,

_a coherent _transverse

single

electron band motion _can

only

be achieved within the characteristic

time scale

~t[j)

_=zi

tj)~

In presence of _a correlation gap whose

amplitude

A~ =

grT~

~ t

[

~ ~,

the formation of electron-hole bound

pairs

and the transverse

single

electron band motion becomes frozen. Interchain virtual

hopping

is

possible however,

and for each

member of the

pair

it _{can occur} within the time scale

Aj

In this way, the _center of _mass the

pair

can move and leads _{to an} effective interchain kinetic

exchange (IEX) coupling. Therefore,

the above

simple

arguments ^{show that the matrix element for} the interchain

pair

transfer will be

proportional

to t

[~~

~/A)

^{for the}

I

or the d directions. Now since the combination of

couplings responsable

^for the

pair

formation _on each chain is

ii

+

#] (see Eq. (25))

the effective IEX

amplitude

at 2

k~

and below

T~

^{will be}

given by J~

~

m 2

cgrv~(#(

₊

#]

_)~

(t

[~~

/Aj )

,

(37)

in lowest order. This form has been confirmed

by

renormalization group calculations

[10]

and from

which,

_c

=

(2

2 Ho

y~)~

~. From

these,

it is also found that the influence of lD

many-body

effects above T~

implies

that the

quantities

that appear in

(37)

are renormalized, lD

self-energy

corrections for

example,

lead _{to a} decrease in the

amplitude

of the transverse

hopping, namely

t

[

~ _~

= zj

(A~ )

t~

~ _~

As we

approach

_T~ from above, the

amplitude

of both g~ ^and g~ are well known to scale to the strong

coupling

sector where

g( _{(A~ )}

₊

g] _{(A~ )}

grvf so

that the bare IEX

amplitude

reduces _to

J~

~

m 2

grv~(2

² Ho

yo)~

(t_[~

/A)) (38)

Therefore, the effective _transverse part ^{of the total} Hamiltonian for energy scales of the order of A~ ^{takes the form}

~i

"

l~ if,,~ °/ (~

+ ~

~F) °j (~

+ ~

~F) (39)

<,.J> _q where

O,,»(q+2k~>=)z*za± «,,(k>«jPa+,p,,(k+2k~+q> _(40>

L ~ «p

N° I NUCLEAR RELAXATION IN ORGANIC CONDUCTORS. ₁₅₃

are the

spin density

_{wave operator near}

2k~

on the _nearest

neighbour

chains and

j.

_«~ are the Pauli matrices

(p

_{= x, y,} z

).

Since

(38)

represents an effective Hamiltonian for energy scales of the order of

A~,

^{it follows that the band} wave vector values k involved in the

definition of the

pair _operators O,

must be such that

[v~~pk k~)[

_~

T~.

In the absence of

single particle

band motion in the transverse direction,

only

effective two-

particle

_{processes are}

possible.

Furthermore below

T~,

t~ can be considered _{as an} irrelevant parameter ^{in the} sense that it has _no influence and it _can therefore be

dropped

from

Ho [10]. H~

now becomes the small

perturbation

of the model. Its influence _on the _transverse

propagation

of AF correlations will be treated in the Random Phase

Approximation (RPA)

while the

purely

ID component ^{of AF} correlations is treated

rigorously. Higher

order

corrections to the RPA in

quasi-

ID systems are

only expected

to emerge in the close

vicinity

of the critical

point,

in the so-called

Ginzburg

critical width which is

sufficiently

small to be

disregarded

for _our purposes. In this scheme of

approximation,

the relevant

quantity X",

in the limit _w

-

0, again obeys

the

following

relation

[10]

Xi(Q+Qo, W>»k(Q+Qo, T>XID(2kF+q, _W), (41)

near the modulation _{wave vector}

Qo

₌

(2 k~,

gr, gr

).

In

RPA,

the 3D

auxiliary susceptibility

near

Qo

is known _to have the

following scaling

form

[10]

k(Q

+

Qo, T)

=

kiD(q

+ 2

kF, T) ii jfi _(qi

+

q(, _{T) xlD(q}

_{+ 2}

_kF, T)j ^{~, (42>}

where

F

~

(q~

+

q[, T>

=

j J~

~

cos

(q~

~

+

q[

~> ⁺

J~

~

cos

(q~

~

+

q[ ~>j/kiD (A~/Eo) ^(43>

The double

pole singularity

of k at

Qo

occurs at

T~,

^which

according

to

(31-33), (38),

and

(42)

reads

T~

= Ap

it _{iy >- ii}

₊

t _(y&

_>-ⁱ

_i-

ⁱ

_ii/Y

,

(44>

where

f

=

(J~~+ J~ )/2

_grv~ and o _~ 8

= +

(J~~

+

J~~> lii~(A~/Eo>i-

ⁱ

xiD(A~>

_~ ⁱ

(45>

is _a constant that takes into account the contribution of correlations to the transition above the

correlation gap

A~.

^Near

T~,

the real part ^{of the total}

susceptibility

at _w =0 and

Q

=

Qo

will present

according

to

(24)

^and

(42)

_a

simple pole singularity X'(Qo, T)

m

x'(A~> (7rv~>~ i(TN/T~

)~^~ l

i iiD(A~/Eo> iii

,

(46>

where _near

T~,

i~~

= 8 y~

j~ _{[(T/T~ )~}

Y

Ii

m y 8

(T T~ )/T~ ⁽⁴⁷⁾

One then _recovers the RPA value

I ₌

,

(48)

for the critical index of the static

staggered susceptibility.

^From

(41-43), (36)

and

(27),

the

relevant ratio

Xi (Q

+

Qo,

w

)/w

for

Tj [q Qo]

and _{w -} 0 takes the form

Xl (Q

+

Q0, °')/°'

_~

(~~ lL(

N

(EF) I~X

ID

(~

_~F,

l~)[~AF

+

f(a

_{~~ +}

f~b _~~b

_~

f~c _~~~]

^~

,

(49)

which is valid for

fo,

_{q, ~} l. For the present ^{model it is found that r} = r

=

w/8 T. The coherence

lengths

_are

given by

~~a

" +

~i (~~, _l~)KID(T) Vi~(~T)~ (50a)

f(~

=

#~~(q(~, T) X(D(2 k~, T) (50b)

tic

=

fli~(q(~, T) xlD(2 kF, T) (5°C)

The

anisotropic

correlation

length

will then be characterized

by

f,

"

to, ~A/~~, (~~)

which

corresponds

to the RPA critical index k

=

1/2

(52)

for the correlation

length.

As for the

dynamical

_exponent I _near

T~,

since

Xi

ⁱⁿ

(49)

scales like

ij/

w at

Qo,

this _means that in order _to get

(48),

_w must scale like

riAF

and _one has

I

=

2

(53)

for the

dynamical

exponent ^{of AF} fluctuations. This value tells _us that the AF order parameter is _a non-conserved

quantity [16].

From

(49)

and

(I),

the AF contribution to

Tj~

_can be written in the form

~

~~

~~

=

~~°

^{~ ~}

~~~~ ~~

_~~

~~~ l~~ to

a

lo

~

/~

~

j

ⁱ ~j

~j/~

~

~~

where

1

~

j~ ^dql j~ ^dql j~ ^{dql Ii}

⁺

^ql~

⁺

^ql~

⁺

^{ql~l~~, (55>}

a -b _-c

with the

integration

limits _a

=

(grT/v~)fo~ij/~~,

b

=

fo~ij/~~

and _c=

io~ij/~~

Two

limiting

_{cases can} then be considered. In the critical

limit,

where T

-

T~, i~~

_-

0+,

_one has

I -

gr~

which leads _to

l~l

(~ Q01

_" t~

(Q0

_~A/~~

,

(~~)

with the coefficient

(Qo) given by

C

(Q0)

_"

~~ Y~

^N

(EF)(~lQo '~ ~N KID(I~N)(l~ f0a fob f0c]T~ ^(5~)

One

immediately verify

from the set of RPA indexes

(48), (52), (53),

and the

scaling expression (7),

that the value

d

= is the

signature

^of 3D critical fluctuations in RPA.

2

N° I NUCLEAR RELAXATION IN ORGANIC CONDUCTORS. 155

In the

opposite

limit where

i~~-I, namely

outside the critical domain where _x,

~ =

with ml

+

~ _F~^q(, X(D(2k~,T)

+.

utside omain

the

4

ffect

(isolatedhains) contribution

T/ [q

2

k~]

= w ~

y(

_N

_{(E~)~ [A~}

^~

_{TkiD (2 k~, T)}

m C

(2 k~ rj/

,

(59)

with if

= y I. This value,

together

^with

(35),

is consistent with the

scaling

relation

(10).

In the presence of _a correlation gap, y = I and

Tj~[q~2k~]

_becomes temperature

independent

in the lD domain. In contrast, for systems ^{in weak}

coupling,

_y is smaller than

unity

and it is non-universal _so that

Tj [q Qo]

decreases with _a downward curvature.

From the two

limiting expressions (56)

and

(58),

_one can

easily verify

that

they satisfy

the extended

scaling hypothesis. Indeed, approaching

the critical domain from above _we find that

the _crossover

scaling

function defined in

(12)

is

given by

~

[Bg~/T~~~]

=

All

B

j~/TY]~

^~~~

,

(60)

where g~ =

j~

is the small

anisotropic

parameter. T~i

(j~

)~~~~~ and ~b~i = y ^are the _two-

particle dimensionality

_crossover temperature ^and

exponent respectively,

B

=

(y8

)~

Tj

and

A

= C

(2 k~)18

~/~[ ar

3 N

(E~) ^fl~ /~

~

f~

~

f~

~

]~

In tum, near TN, ^{the critical}

expression (56)

_can be written in the

scaling

form

(13)

with the non,universal _constant of

(15) given by

d~

=

Ay

^*

T)

,

(61)

where A is evaluated _at

T~.

The critical width

At~m (Tn- T~)/T~ giving

the range ⁱⁿ

temperature ^above

T~

where the critical contribution

(56)

exceeds the

paramagnetic

one

(58).

When the _two

expressions

are of the _same order of

magnitude

_one gets

Atfl

₌

7r~(32 y8 (7rTN~~F) f0a fob f0cl ^(~~)

For weak transverse

anisotropy, to

~,~

s I and for strong

umklapp coupling

it therefore leads

to

Atn~

I which is

large.

Otherwise, for

i~~» i~~

_transverse correlations should grow

preferentially

in _{one transverse} direction and the _{crossover can} first be _{seen as} a ID _to 2D

crossover so that

Atn

^{will further increase. In}

2D,

_one has

according

to

(48), (52-53)

and

(7),

if

= I which leads _{to a} much stronger temperature

dependence

^for

Tj

_near the 2D critical

point

determined

by

the

expression (44) by putting J~

=

0.

4. Itinerant

quasi-one,dimensional antiferromagnets.

In the absence of _a correlation gap and if the intrachain

couplings

_are not too

small,

the IEX

coupling

still drives the AF

phase

transition. The

elementary

IEX

amplitude

becomes energy scale

dependent

whenever A~ «

T~.

Indeed, in the weak intrachain

coupling domain,

electron-